We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) by introducing a unifying abstract framework, and derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods.

翻译：我们考虑具有无限时滞的非线性时滞微分方程与更新方程。通过引入统一的抽象框架，我们在Gyllenberg等人(Appl. Math. Comput., 2018)的基础上拓展研究，并利用伪谱离散化方法推导出有限维逼近系统。针对更新方程，我们通过积分将其在绝对连续函数空间中进行重构。证明了原始方程与其逼近系统之间平衡点的一一对应关系，同时验证了线性化与离散化具有可交换性。最重要的成果是：当选取拉盖尔多项式的缩放零点或极值点作为配点节点时，证明了线性(化)方程伪谱逼近的特征根收敛性。这一性质确保了在逼近系统维数足够大的情况下，有限维系统能够正确复现原始线性方程的稳定性特征。通过多项数值实验验证了该结论，实验同时展示了该方法对非线性方程平衡点分岔分析的有效性。证明收敛性的新方法还揭示了拉盖尔零点与极值点对应微分矩阵谱的精确位置，为伪谱方法求解微分方程的数值特性提供了新的理论见解。